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Ⅲ Cyclic Code. • shift registers with feedback connections inherent algebraic structure  Algorithmic technique • random-error correction burst-error correction. Def : Cyclic Code (i) one linear code ( subset of GF(q)[x]/x n -1 )

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•shift registers with feedback connections

inherent algebraic structure  Algorithmic technique

• random-error correction burst-error correction

Def : Cyclic Code (i) one linear code ( subset of GF(q)[x]/xn-1 )

(n, k) (ii) x • c(x) mod xn-1 C(x) if c(x) C(x)

Shift right i 

Shift left n-i

• v = (v0, v1, ..., vn-1) : code vector

v(1) = (vn-1, v0, v1, …, vn-2)

v(i) = (vn-i, vn-i+1, …, vn-1, v0, v1, …, vn-i-1)

• v(x) = v0 + v1x + … + vn-1xn-1 : code polynomial

• v(i)(x) = xiv(x)/xn-1

Yuh-Ming Huang, CSIE NCNU

Consider : g(x), xg(x), x2g(x), …, xn-r-1g(x)

=

=

=

code polynomials

g(1)(x)

g(2)(x)

g(n-r-1)(x)

Def : generator polynomial : The nonzero monic code polynomial of

minimum degree in a cyclic code C

g(x) = g0 + g1x + … + gr-1xr-1 + xr

( note : multiply by a field element to make it monic )

• g(x) is unique

• In GF(2), g0 = 1

We claim : r = n - k

For (n, k) cyclic code

v(x) = u0g(x) + u1xg(x) + u2x2g(x) + … + un-r-1xn-r-1g(x)

= ( u0 + u1x + u2x2 + … + un-r-1xn-r-1 )g(x)

= u(x)g(x) is also a code polynomial

Note If r = n–k, deg u(x) = k-1 u(x) : information polynomial

Yuh-Ming Huang, CSIE NCNU

Thm : Let g(x) = 1 + g1(x) + … + gr-1xr-1 + xr be the generator polynomial

in an (n, k) cyclic code. A binary polynomial of degree n-1 or less

is a code polynomial iff it is a multiple of g(x).

Proof:

•  v(x) = a(x)g(x) + b(x) ∴ b(x) = v(x) + a(x)g(x) and deg b(x)<deg g(x)

•  v(x) = (a0+a1x +…+an-r-1xn-r-1)g(x)= a0g(x)+ a1xg(x)+…+an-r-1xn-r-1g(x)

# of binary polynomial with degree ≤ n-1 that are multiples of g(x) = 2n-r

2n-r = 2k  r = n-k

Thm : In an (n,k) cyclic code, there exists one and only one code

polynomial of degree n-k

g(x) = 1 + g1x + g2x2 + … + gn-k-1xn-k-1 + xn-k

Every code polynomial is a multiple of g(x) and every binary polynomial of degree n-1 or less that is a multiple of g(x) is a code polynomial.

Q:g(x), xg(x), x2g(x), …, xk-1g(x) are linear independent ?

Yuh-Ming Huang, CSIE NCNU

Thm : g(x) | xn+1 (xn-1 )

pf: xn – 1 = Q(x)g(x) + s(x)  兩邊 mod xn-1

 0 = Q(x)g(x)/xn-1 + s(x)

Q:

Thm : If g(x) is a polynomial of degree n-k and is a factor of xn+1(xn-1),

then g(x) generate an (n, k) cyclic code

pf: (1) v(x) = u0 g(x)+u1xg(x)+…+uk-1xk-1g(x)

= (u0 +u1x+…+uk-1xk-1)g(x)= v0+v1x+…+vn-1xn-1

there exists 2k code polynomials  (n,k) linear code

(2) If v(x) is a code polynomial, then xv(x) = vn-1(xn-1)+v(1)(x)

 v(1)(x) is a multiple of g(x)  v(1)(x) is also a code polynomial

Yuh-Ming Huang, CSIE NCNU

Q: For large n, xn+1 may have many factors of degree n-k.

How to select g(x)?

Non-Systematic Encoding

 v(x) = u(x)g(x)

• xn-ku(x) =a(x)g(x)+ t(x)

• t(x)+ xn-ku(x) = a(x)g(x)

& deg[t(x)+ xn-ku(x)]  n-1

∴ It’s a code polynomial

Systematic Encoding

v(x) = xn-ku(x) + t(x)

where t(x) is chosen st. (i) deg t(x) < n-k

(ii) v(x)/g(x) = 0

∴ t(x) = - xn-ku(x)/g(x)

 t0 + t1x + … + tn-k-1xn-k-1

v =(t0, t1, …, tn-k-1, u0, u1, …,uk-1)

Q: t(x) physical mean?

Yuh-Ming Huang, CSIE NCNU

Messages

Code Vectors

Code polynomials

( 0 0 0 0 )

0 0 0 0 0 0 0

0 = 0•g(x)

( 1 0 0 0 )

1 1 0 1 0 0 0

1+x+x3 = 1•g(x)

( 0 1 0 0 )

0 1 1 0 1 0 0

x+x2+x4 = x•g(x)

( 1 1 0 0 )

1 0 1 1 1 0 0

1+x2+x3+x4 = (1+x)•g(x)

( 0 0 1 0 )

0 0 1 1 0 1 0

x2+x3+x5 = x2•g(x)

( 1 0 1 0 )

1 1 1 0 0 1 0

1+x+x2+x5 = (1+x2)•g(x)

( 0 1 1 0 )

0 1 0 1 1 1 0

x+x3+x4+x5 = (x+x2)•g(x)

( 1 1 1 0 )

1 0 0 0 1 1 0

1+x4+x5 = (1+x+x2)•g(x)

( 0 0 0 1 )

0 0 0 1 1 0 1

x3+x4+x6 = x3•g(x)

( 1 0 0 1 )

1 1 0 0 1 0 1

1+x+x4+x6 = x3•g(x)

( 0 1 0 1 )

0 1 1 1 0 0 1

x+x2+x3+x6 = (x+x3)•g(x)

( 1 1 0 1 )

1 0 1 0 0 0 1

1+x2+x6 = (1+x+x3)•g(x)

( 0 0 1 1 )

0 0 1 0 1 1 1

x2+x4+x5+x6 = (x2+x3)•g(x)

( 1 0 1 1 )

1 1 1 1 1 1 1

1+x+x2+x3+x4+x5+x6 = (1+x2+x4)•g(x)

( 0 1 1 1 )

0 1 0 0 0 1 1

x+x5+x6 = (x+x2+x3)•g(x)

( 1 1 1 1 )

1 0 0 1 0 1 1

1+x3+x5+x6 = (1+x+x2+x3)•g(x)

non-systematic

Yuh-Ming Huang, CSIE NCNU

Messages

Code Vectors

Code polynomials

( 0 0 0 0 )

0 0 0 0 0 0 0

0 = 0•g(x)

( 1 0 0 0 )

1 1 0 1 0 0 0

1+x+x3 = 1•g(x)

( 0 1 0 0 )

0 1 1 0 1 0 0

x+x2+x4 = x•g(x)

( 1 1 0 0 )

1 0 1 1 1 0 0

1+x2+x3+x4 = (1+x)•g(x)

( 0 0 1 0 )

1 1 1 0 0 1 0

1+x+x2+x5 = (1+x2)•g(x)

( 1 0 1 0 )

0 0 1 1 0 1 0

x2+ x3+x5 = x2•g(x)

( 0 1 1 0 )

1 0 0 0 1 1 0

1+x4+x5 = (1+x+x2)•g(x)

( 1 1 1 0 )

0 1 0 1 1 1 0

x+x3+x4+x5 = (x+x2)•g(x)

( 0 0 0 1 )

1 0 1 0 0 0 1

1+x2+x6 = (1+x+x3)•g(x)

( 1 0 0 1 )

0 1 1 1 0 0 1

x+x2+x3+x6 = (x+x3 )•g(x)

( 0 1 0 1 )

1 1 0 0 1 0 1

1+x+x4+x6 = (1+x3)•g(x)

( 1 1 0 1 )

0 0 0 1 1 0 1

x3+x4+x6 = x3•g(x)

( 0 0 1 1 )

0 1 0 0 0 1 1

x+x5+x6 = (x+x2+x3)•g(x)

( 1 0 1 1 )

1 0 0 1 0 1 1

1+x3+x5+x6 = (1+x+x2+x3)•g(x)

( 0 1 1 1 )

0 0 1 0 1 1 1

x2+ x4+x5+x6 = (x2+x3)•g(x)

( 1 1 1 1 )

1 1 1 1 1 1 1

1+x+x2+x3+x4+x5+x6 = (1+x2+x5)•g(x)

systematic

X3(x2+1)

X3+x+1)

= x2

Yuh-Ming Huang, CSIE NCNU

k x n

Generator & Parity-check Matrices

(一)

v = u • G

v(x) = (u0 + u1x + … + uk-1xk-1)g(x)

Q: GHT=0

xn+1 = g(x)h(x)

where h(x) = h0 + h1x + … + hkxk : parity polynomial

xkh(x-1) = hk + hk-1x + … + h0xk

Q:

∵ xkh(x-1) | xn+1 ∴ xkh(x-1) generate an (n, n-k) cyclic code

Yuh-Ming Huang, CSIE NCNU

(n-k) x n

Note If v(x) = a(x)g(x)

v(x)h(x) = a(x)g(x)h(x) = a(x)(xn+1) = a(x) + xna(x)

∵ deg a(x) ≤ k-1

0 1 …k-1 n n+1 …n+k-1

deg = j 項係數

Yuh-Ming Huang, CSIE NCNU

(n-k) x n

(二)

consider each i corresponding to an information place

xn-k+i = ai(x)g(x) + bi(x) 0 ≤ i ≤ k-1 xn-k, xn-k+1, …, xn-1

where bi(x) = bi0 + bi1x + … + bi,n-k-1xn-k-1

 bi(x) + xn-k+i are code polynomials

Q:

(1) linear independent?

(2) 與

xn-ku(x)/g(x)之關係?

G =

k x n

(3) check VHT=0

(4) (一)G與(二)G之關係?

Yuh-Ming Huang, CSIE NCNU

xn-ku(x) = u0xn-k + u1xn-k+1 + … + uk-1xn-1

g(x) g(x)

= u0b0(x) + u1b1(x) + … + uk-1bk-1(x)

= t0 + t1x + … + tn-k-1xn-k-1

Ans (3)

V = (v0,v1,…,vn-k-1,vn-k,vn-k+1,…,vn-1)

= (t0,t1,…,tn-k-1,u0,u1,…,uk-1)

VHT=V0+ Vn-kb00+ Vn-k+1b10+…+ Vn-1bk-1,0

= t0+ u0b00+ u1b10+…+ uk-1bk-1,0

Yuh-Ming Huang, CSIE NCNU

g0

g1

-1

=

=

B

A

(一). b(x) mod g(x)

g(x) = x2 + g1x + g0

b(x) = b3x3 + b2x2 + b1x + b0

n-k = deg[g(x)] = 2

Ax + B (=s(x))

b0,b1,b2,b3

• b3在第3clock 啟動前位於 -1 下方

• i.e.就b3x3單獨而言,在第4clock 後,registers內容為b3x3 mod g(x)

G

Yuh-Ming Huang, CSIE NCNU

g0

g1

-1

(二).

x2b(x) mod g(x)

• Q:

• xb(x)/g(x)

• (i) while complete the computation

• b(x)/g(x)  s(x)

• (ii) G disable

• (iii) one right shift

• (iv) result = xs(x)/g(x)

• xb(x)  input 0, b0, b1, b2 , b3

• (2) x2b(x)/g(x)

• input 0, 0, b0, b1, b2 , b3

b0,b1,b2,b3

• 就b3x3單獨而言, 在第4 clock 結束後, registers內容為b3x5 mod g(x). i.e. 相當於 (一), b3由左方輸入但需 6個clock.

• 同理 for b2, b1, b0

Yuh-Ming Huang, CSIE NCNU

g2

g1

gn-k-1

b0

b1

b2

bn-k-1

Message Xn-ku(X)

Code word

Parity-check

digits

Encoding

< Method Ⅰ>

step (i) xn-ku(x)

(ii) xn-ku(x)/g(x) = b(x)

(iii) b(x) + xn-ku(x)

n-k stage

Figure 4.1 Encoding circuit for an (n, k) cyclic code with generator polynomial

g(X) = 1 + g1X + g2X2 + … + gn-k-1Xn-k-1 + Xn-k

Yuh-Ming Huang, CSIE NCNU

Code word

Parity digits

Message Xn-ku(X)

Example 4.5

Consider the (7, 4) cyclic code generated by g(X) = 1 + X + X3. The encoding circuit based on g(X) is shown in Figure 4.2. Suppose that the message u = (1 0 1 1) is to be encoded. As the message digits are shifted into the register, the contents in the register are as follows:

After four shifts, the contents of the register are (1 0 0). Thus, the complete code vector is (1 0 01 0 1 1) and the code polynomial is 1 + X3 + X5 + X6.

(1 0 1 1)

Figure 4.2 Encoder for the (7, 4) cyclic code generated by g(X) = 1 + X + X3

Yuh-Ming Huang, CSIE NCNU

--- (1)

given k information digits (vn-k, vn-k+1, …, vn-1) = (u0, u1, … , uk-1)

By (1), to determine the n-k parity-check digits (v0, v1, … , vn-k-1)

vn-k-1 = h0vn-1 + h1vn-2 + … + hk-1vn-k = uk-1 + h1uk-2 + … + hk-1u0

vn-k-2 = h0vn-2 + h1vn-3 + … + hk-1vn-k-1 = uk-2 + h1uk-3 + … + hk-2u0 + hk-1vn-k-1

+

+

+

+

Gate2

hk-1

hk-2

h2

h1

u0

u1

• • •

Uk-2

Uk-1

Gate1

° Output to channel

k stage

Figure 4.3 Encoding circuit for an (n, k) cyclic code based on the parity

polynomial h(x) = 1 + h1X + h2X2 + …+ hk-1Xk-1+Xk

Yuh-Ming Huang, CSIE NCNU

Input

Gate 1

p

Output

h0 = 1

Figure 4.4 Encoding circuit for the (7, 4) cyclic code based on its parity

polynomial h(x) = 1 + X + X2 + X4

Suppose that the message to be encoded is (1 0 1 1).

Then v3 = 1, v4 = 0, v5 = 1, v6 = 1. The first parity-check digit is

v2 = v6 + v5 + v4 = 1 + 1 + 0 = 0

The second parity-check digit is

Q: 觀察 Method Ⅰ,Ⅱ 結果相同

v1 = v5 + v4 + v3 = 1 + 0 + 1 = 0

The third parity-check digit is

v0 = v4 + v3 + v2 = 0 + 1 + 0 = 1

Thus, the code vector that corresponds to the message (1 0 1 1) is

(1 0 0 1 0 1 1).

Yuh-Ming Huang, CSIE NCNU

g2

g1

gn-k-1

sn-k-1

r(X)

s0

s1

vector

Syndrome Computation & Error Detection

< Type Ⅰ>

r(x) = r0 + r1x + … + rn-1xn-1

= a(x)•g(x) + s(x) deg s(x) ≤ n-k-1

r(x) is a code polynomial

s(x) = 0

syndrome

r(x)/g(x) ≈

Figure 4.5 An (n-k)-stage syndrome circuit with input from the left end.

Yuh-Ming Huang, CSIE NCNU

Then s(1)(x) ( = x• s(x)/g(x) ) is the syndrome of

r(1)(x) (= x • r(x)/xn+1)

pf:

Yuh-Ming Huang, CSIE NCNU

r = (r0, r1, … ,rn-k-1, rn-k, … , rn-1) = (t0, t1, … , tn-k-1, u0, u1, … ,uk-1)

r•HT = (s0, s1, … , sn-k-1)

s0 = r0 + rn-kb00 + rn-k+1b10 + … + rn-1bk-1,0 = r0 + t0

s1 = r1 + rn-kb01 + rn-k+1b11 + … + rn-1bk-1,1 = r1 + t1

sn-k-1 = rn-k-1 + rn-kb0, n-k-1 + rn-k+1b1, n-k-1 + … + rn-1bk-1, n-k-1

Syndrome is simply the vector sum of the received parity digits and the parity-check digits recomputed from the received information digits

∵ r(x) = r0 + r1x + … + rn-k-1xn-k-1 + xn-k•u(x)

u(x) = u0 + u1x + … + uk-1xk-1

∴ r(x)/g(x) = r0 + r1x + … + rn-k-1xn-k-1 + xn-k•u(x)/g(x)

=

t0 + t1x + … + tn-k-1xn-k-1

( rn-k, rn-k+1, … , rn-1 )G = ( t0, t1, … , tn-k-1, rn-k, rn-k+1, rn-1 )

Yuh-Ming Huang, CSIE NCNU

A syndrome circuit for the (7, 4) cyclic code generated by g(X) = 1 + X + X3 is shown in Figure 4.6. Suppose that the receive vector is r = (0 0 1 0 1 1 0). The syndrome of r is s = (1 0 1). As the received vector is shifted into the circuit, the contents in the register are given in Table 4.3.

Gate

input

Gate

1

1

0

Figure 4.6 Syndrome circuit for the (7, 4) cyclic code generated by g(X) = 1 + X + X3.

Table 4.3 CONTENTS OF THE SYNDROME REGISTER SHOWN IN FIGURE

4.6 WITH r = (0 0 1 0 1 1 0) AS INPUT

Yuh-Ming Huang, CSIE NCNU

< Type Ⅱ>

xn-kr(x)/g(x) ≈

g2

g1

gn-k-1

sn-k-1

s0

s1

r(X)

Figure 4.7 An (n-k)-stage syndrome circuit with

input from the right end.

Cor : The contents of the registers form the syndrome s(n-k)(x) of r(n-k)(x),

which is the (n-k)th cyclic shift of r(x)

Note :

• xn-kr(x)/g(x) = [xn-kr(x)/xn+1]/g(x) = r(n-k)(x)/g(x) = s(n-k)(x)

• r(x) = v(x) + e(x)∴ s(x) = r(x)/g(x) = e(x)/g(x)

xn-kr(x)

= Q(x)(xn+1) + R(x)

= Q(x)p(x)g(x) + q(x)g(x) + s(x)

= [Q(x)p(x) + q(x)]g(x) + s(x)

Yuh-Ming Huang, CSIE NCNU

Decoding

Meggitt decoder

received digits are decoded one at a time and

each digit is decoded with the same circuitry

<see next page>

Note : If Syndrome register ≠ 0’s at the end of decoding ( n shifts )

uncorrectable error pattern

syndrome

r(x)

r(1)(x)

r1(x)

r1(1)(x)

s(x)

s(1)(x)

s1(x)

s1(1)(x)

• r1(x) = r(x) + xn-1

• s1(X) = s(x) + xn-1/g(x)

• s1(1)(x) = x•s1(x)/g(x) = xs(x)/g(x) + [x•xn-1/g(x)]/g(x) = xs(x)/g(x) + xn/g(x) = s(1)(x) + 1

xn-1 = Q(x)g(x)

xn = Q(x)g(x) + 1

Yuh-Ming Huang, CSIE NCNU

Corrected

vector

ri

r(X)

Buffer registers

Gate

vector

Feedback connection

Gate

Gate

Syndrome register

ei

Gate

Error pattern detection circuit

Syndrome modification

< Type Ⅰ>

(r0, r1, … , rn-1)

=

s(x) --- r(x)

s(1)(x) --- r(1)(x)

Figure 4.8 General cyclic code decoder with received polynomial r(X)

shifted into the syndrome register from the left end

Yuh-Ming Huang, CSIE NCNU

Error pattern

e(X)

Syndrome

s(X)

Syndrome vector

(s0, s1, s2)

e6(X) = X6

e5(X) = X5

e4(X) = X4

e3(X) = X3

e2(X) = X2

e1(X) = X1

e0(X) = X0

s(X) = 1 + X2

s(X) = 1 + X + X2

s(X) = X + X2

s(X) = 1 + X

s(X) = X2

s(X) = X

s(X) = 1

(1 0 1)

(1 1 1)

(0 1 1)

(1 1 0)

(0 0 1)

(0 1 0)

(1 0 0)

Consider the decoding of the (7, 4) cyclic code generated by g(X)= 1 + X + X3. This code has minimum distance 3 and is capable of correcting any single error over a block of seven digits. There are seven single-error patterns. These seven error patterns and the all-zero vector form all the coset leaders of the decoding table. Thus, they form all the correctable error patterns. Suppose that the received polynomial r(X) = r0 + r1X + r2X2 + r3X3 + r4X4 + r5X5 + r6X6 is shifted into the syndrome register from the left end. The seven single-error patterns and their corresponding syndromes are listed in Table 4.4.

TABLE 4.4 ERROR PATTERNS AND THEIR SYNDROMES WITH THE RECEIVED POLYNOMIAL r(x) SHIFTED INTO THE SYNDROME REGISTER FROM THE LEFT END

Yuh-Ming Huang, CSIE NCNU

r’(X)

r(X)

Multiplexer

Output

Input

Gate

Gate

0

0

1

Gate

Figure 4.9 Decoding circuit for the (7, 4) cyclic code generated by

g(X) = 1 + X + X3.

Yuh-Ming Huang, CSIE NCNU

0

1

1

1

1

1

1

0

0

1

1

1

1

0

1

1

1

0

1

0

0

1

1

1

1

1

0

1

0

0

0

1

1

0

0

1

1

1

0

1

0

1

1

Syndrome

register

Buffer register

Correction

Pointer

Initial

1st shift

2st shift

3rd shift

Yuh-Ming Huang, CSIE NCNU

1

1

0

0

0

1

0

0

0

1

1

0

1

1

1

0

1

1

1

0

1

1

1

1

1

0

1

Error corrected

0

0

1

0

Corrected word

1

0

0

0

0

0

0

0

0

1

0

0

4th shift

5th shift

6th shift

7th shift

Figure 4.10 Error-correction process of the circuit shown in Figure 4.9

Yuh-Ming Huang, CSIE NCNU

ri

r(X)

Buffer register r(X)

Gate

vector

Corrected

vector

Gate

Feedback connection

Gate

Syndrome register

(Syndrome

modification)

Gate

ei

Error-pattern detection circuit

• s(n-k)(x) : syndrome of r(n-k)(x)

• If en-1 = 1 In r(n-k)(x), the digit rn-1 is at the location xn-k-1r1(n-k)(x) = r(n-k)(x) + xn-k-1

•  s1(n-k)(x) = s(n-k)(x) + xn-k-1

< Type Ⅱ>

(r0, r1, … ,rn-1)

=

Note:

<Type I> circular shift 後考量

<Type II> circular shift 前考量

Figure 4.11 General cyclic code decoder with received polynomial r(X) shifted into the syndrome register from the right end.

Yuh-Ming Huang, CSIE NCNU

Again, we consider the decoding of the (7, 4) cyclic code generated by g(X) = 1 + X + X3. Suppose that the received polynomial r(X) is shifted into the syndrome register from the right end. The seven single-error patterns and their corresponding syndromes are listed in Table 4.5.

We see that only when e(X) = X6 occurs, the syndrome is (0 0 1) after the entire received polynomial r(X) has been shifted into the syndrome register. If the single error occurs at the location Xi with i ≠ 6, the syndrome in the register will not be (0 0 1) after the entire received polynomial r(X) has been shifted into the syndrome register. However, another 6 – i shifts, the syndrome register will contain (0 0 1). Based on this fact, we obtain another decoding circuit for the (7, 4) cyclic code generated by g(X) = 1 + X + X3, as shown in Figure 4.12. We see that the circuit shown in Figure 4.9 and the circuit shown in Figure 4.12 have the same complexity.

Yuh-Ming Huang, CSIE NCNU

e(X)

Syndrome

s(3)(X)

Syndrome vector

(s0, s1, s2)

e(X) = X6

e(X) = X5

e(X) = X4

e(X) = X3

e(X) = X2

e(X) = X1

e(X) = X0

s(3)(X) = X2

s(3)(X) = X

s(3)(X) = 1

s(3)(X) = 1 + X2

s(3)(X) = 1 + X + X2 s(3)(X) = X + X2

s(3)(X) = 1 + X

(0 0 1)

(0 1 0)

(1 0 0)

(1 0 1)

(1 1 1)

(0 1 1)

(1 1 0)

TABLE 4.5 ERROR PATTERNS AND THEIR SYNDROMES WITH THE RECEIVED POLYNOMIAL r(x) SHIFTED INTO THE SYNDROME REGISTER FROM THE RIGHT END

Yuh-Ming Huang, CSIE NCNU

r(X)

r’(X)

Multiplexer

Output

Input

Gate

Gate

Gate

Figure 4.12 Decoding circuit for the (7, 4) cyclic code generated by

g(X) = 1 + X + X3.

Yuh-Ming Huang, CSIE NCNU